Appendix

  We present a proof of correctness of the signal mapping algorithm detailed in Section 2.3. We use the same notation introduced in Section 2.3.

Claim: This signal layer mapping meets the following constraints:

1. The network channel limits are observed.
2. No source S_k transmits more than its total share, w_kB.
3. w_k >= w_j implies that A_k,n <= A_j,n forall n

Proof: In the proof we refer to line numbers in the algorithm detailed in Figure 6.

1. represents the unused bandwidth available in network channel c. Thus, the check in line 15 guarantees that no source will cause the rate limit of a network channel to be exceeded
2. represents the portion of the bandwidth share of source S_l that has not been utilized by the source. Thus, the decrementing of in line 21, along with the check in line 12, guarantees that no source will transmit more that its total share, which is w_lB.
3. The algorithm iterates through the average weight list looking for the largest remaining weight, . Now assume that for the two sources in question S_k, and S_j, all signal layers 1,..., n-1 have been mapped to network channels. Next assume that the result of the algorithm's output for layer n is such that mu_k,n > mu_j,n

   This along with line 9 imply that

   

   Line 21 of the algorithm implies that in order for (1) to hold the following must also be true:

   

   which implies that

   w_k < w_l

   Finally, the arg max directive in line 9 furnishes us with the bootstrap for the induction in the first iteration which completes the proof of statement 3 of the claim. QED